Research

My research is on the interface of dynamical systems, geometry, topology, group theory, and combinatorics. Its main focus is on the combinatorial, geometric, algebraic, and algorithmic aspects of holomorphic dynamical systems and self-similar group theory.

Most recently, I have been working on Thurston's theory of rational maps and on fundamental connections between holomorpic dynamics and geometric group theory provided by Sullivan’s dictionary and the theory of iterated monodromy groups (IMGs). Specifically, I investigate how topological an geometric complexity of Julia sets of rational maps (e.g., Ahlfors regular conformal dimension) and algebraic complexity of the corresponding IMGs (e.g., growth and amenability) are encoded by dynamical properties of the maps.

A detailed description of my past work may be found in my
 Research Statement. 

The figure indicates connections that I explore in my research. Pictures courtesy of C. Bishop, D. Calegary, and C. McMullen

Thurston's characterization of rational maps

To better understand dynamical properties of rational maps, it is often convenient to abstract from the rigid complex structure and consider the more general case of topological branched covering maps. One of the celebrated theorems of Thurston characterizes postcritically-finite (in short, pcf) branched covers of the topological 2-sphere that can be realized by rational maps. Even though it is one of the most important and influential results in holomorphic dynamics, it still has only few applications. In my research I seek new insights and further applications of Thurston's characterization theorem, for instance, in the classification theory of rational maps.


Relevant work and talks

Blowing up a Lattès map: if there is at least one horizontal and at least one vertical flap, we obtain a rational map

Combinatorial models of rational maps and their applications

By far the most studied and well understood class of maps in holomorphic dynamics is the family of polynomials. In the 1980's, Douady and Hubbard developed various combinatorial models for pcf polynomials. A combinatorial model is a finite certificate (which may be of very different formats, for example, a finite sequence of rational numbers or a finite graph) that is assigned to each map within the class. Later these models were used to completely classify all pcf polynomials in terms of finite data. However, the case of general rational maps is much more complicated and still draws lots of attention. One of the ultimate goals of my research is to construct "good" combinatorial models for all pcf rational maps and use these models to classify all pcf rational maps. 

In fact, combinatorial models are also quite useful for investigation of other long-standing questions in holomorphic dynamics. For instance, they help to understand the dynamical behavior of simple closed curves under the pullback by a rational map, which in its turn provides new insights on Thurston's characterization theorem. Furthermore, combinatorial models are very handy in algebraic and algorithmic aspects of holomorphic dynamical systems, which are addressed below.


Relevant work and talks

Iterative construction of an invariant graph.


An invariant tree for a Lattès map.
Pictures courtesy of D. Meyer

Iterated monodromy groups and their properties 

The recent and rapidly developing theory of iterated monodromy groups (in short, IMGs) provides a natural bridge between dynamical systems and geometric group theory. IMGs were introduced by Nekrashevych in 2001 as self-similar groups that are naturally associated to pcf branched covering maps on the 2-sphere. Since then, they were used to solve many important problems in holomorphic dynamics, including the Hubbard twisted rabbit problem and (some instances of) the global curve attractor problem. Furthermore, IMGs frequently provide examples of groups with interesting properties that are "exotic" from the point of view of classical group theory, such as groups of intermediate growth and amenable groups of exponential growth. However, properties of the IMGs of general pcf rational maps are still poorly understood and require a systematic study. 


The main goal in this research direction is to study algebraic properties (such as torsion, growth, and amenability) of the IMGs of pcf branched covering maps and of pcf rational maps in particular. Such investigations may not only help to better understand dynamical systems but also deliver new methods for group theory.


Relevant work and talks

Caption
Proving exponential growth of an IMG using tilings 

Algorithmic aspects

Nekrashevych's theory of IMGs and bisets allows to develop computational techniques to study properties of pcf rational maps. The main protagonists in this area are Bartholdi, Dudko, and Nekrashevych who designed a variety of symbolic algorithms for pcf branched covers. Besides, mapping class groups techniques were recently used to provide a new solution to the Hubbard twisted rabbit problem. I aim to further explore connections between dynamical and group theoretical concepts, and develop efficient algorithms that analyze pcf rational maps using their combinatorial models.


Relevant work and talks

Crochet Algrothim: Decomposing a rational map into pieces based on the topological structure of the Julia set